#2085132

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,11)(8,9)(10,12)(14,16)(15,19)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36);(5,6,9,8)(7,10,12,11)(13,14,17,16)(15,18,19,20)(23,27)(24,30)(31,35)(32,36);(6,8)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,18,19,20)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fg,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1992328);